Utilizing multiple modalities to learn a good distance metric is of vital importance for various clinical applications. However, it is common that modalities are incomplete for some patients due to various technical and practical reasons in healthcare datasets. Existing metric learning methods cannot directly learn the distance metric on such data with missing modalities. Nevertheless, the incomplete data contains valuable information to characterize patient similarity and modality relationships, and they should not be ignored during the learning process. To tackle the aforementioned challenges, we propose a metric learning framework to perform missing modality completion and multi-modal metric learning simultaneously. Employing the generative adversarial networks, we incorporate both complete and incomplete data to learn the mapping relationship between modalities. After completing the missing modalities, we use the nonlinear representations extracted by the discriminator to learn the distance metric among patients. Through jointly training the adversarial generation part and metric learning, the similarity among patients can be learned on data with missing modalities. Experimental results show that the proposed framework learns more accurate distance metric on real-world healthcare datasets with incomplete modalities, comparing with the state-of-the-art approaches. Meanwhile, the quality of the generated modalities can be preserved.